Matchings under One-Sided Preferences with Soft Quotas

Abstract

Assigning applicants to posts in the presence of the preferences of applicants and quotas associated with posts is extensively investigated. For a post, lower quota guarantees, and upper quota limits the number of applicants assigned to it. Typically, quotas are assumed to be fixed, which need not be the case in practice. We address this by introducing a soft quota setting, in which every post is associated with two values – lower target and upper target which together denote a range for the intended number of applicants in any assignment. Unlike the fixed quota setting, we allow the number of applicants assigned to a post to fall outside the range. This leads to assignments with deviation. Here, we study the problem of computing an assignment that has two orthogonal optimization objectives – minimizing the deviation (maximum or total) w.r.t. soft quotas and ensuring optimality w.r.t. preferences of applicants (rank-maximality or fairness). The order in which these objectives are considered, the different possibilities to optimize deviation combined with the well-studied notions of optimality w.r.t. preferences open up a range of optimization problems of practical importance. We present efficient algorithms based on flow-networks to solve these optimization problems.